Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes

Abstract

Consider the number of faces arising from unimodular triangulations of random polytopes associated with Erdős--Rényi graphs. Focusing on two-dimensional faces of regular unimodular triangulations of random symmetric edge polytopes, we derive precise asymptotic expansions for the expectation and variance and then establish a quantitative central limit theorem with an explicit Berry--Esseen bound. The proof combines an algebraically derived characterization of faces in terms of local graph configurations with a detailed combinatorial--geometric study of the relevant configurations. The normal approximation is established using the discrete Malliavin--Stein method. We also determine the asymptotic orders of the expectation and variance and establish a quantitative central limit theorem for the number of faces in each fixed higher dimension. These results identify a common local mechanism underlying the fluctuations of face numbers and extend the distributional theory of random symmetric edge polytopes [Donzelmann et al., arXiv:2603.10650] beyond edge statistics.

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BibTeX RIS

BibTeX

@misc{su2026berryesseen,
  title = {{Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes}},
  author = {Zhonggen Su and Huilian Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28088},
  eprint = {2610.28088},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28088},
}

RIS

TY  - UNPB
TI  - Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes
AU  - Zhonggen Su
AU  - Huilian Zheng
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28088
UR  - https://arxiv.org/abs/2610.28088
AB  - Consider the number of faces arising from unimodular triangulations of random polytopes associated with Erdős--Rényi graphs. Focusing on two-dimensional faces of regular unimodular triangulations of random symmetric edge polytopes, we derive precise asymptotic expansions for the expectation and variance and then establish a quantitative central limit theorem with an explicit Berry--Esseen bound. The proof combines an algebraically derived characterization of faces in terms of local graph configurations with a detailed combinatorial--geometric study of the relevant configurations. The normal approximation is established using the discrete Malliavin--Stein method. We also determine the asymptotic orders of the expectation and variance and establish a quantitative central limit theorem for the number of faces in each fixed higher dimension. These results identify a common local mechanism underlying the fluctuations of face numbers and extend the distributional theory of random symmetric edge polytopes [Donzelmann et al., arXiv:2603.10650] beyond edge statistics.
ER  -

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